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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvat | Structured version Visualization version GIF version |
Description: If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 29797 analog.) (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
lcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
icvat.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcvat.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lcvat.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lcvat.l | ⊢ (𝜑 → 𝑇𝐶𝑈) |
Ref | Expression |
---|---|
lcvat | ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊕ 𝑞) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
3 | lcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
4 | lcvat.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lcvat.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
6 | lcvat.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | icvat.c | . . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
8 | lcvat.l | . . . 4 ⊢ (𝜑 → 𝑇𝐶𝑈) | |
9 | 1, 7, 4, 5, 6, 8 | lcvpss 35178 | . . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
10 | 1, 2, 3, 4, 5, 6, 9 | lrelat 35168 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
11 | 4 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑊 ∈ LMod) |
12 | 5 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑇 ∈ 𝑆) |
13 | 6 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑈 ∈ 𝑆) |
14 | simp2 1128 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑞 ∈ 𝐴) | |
15 | 1, 3, 11, 14 | lsatlssel 35151 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑞 ∈ 𝑆) |
16 | 1, 2 | lsmcl 19478 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆) → (𝑇 ⊕ 𝑞) ∈ 𝑆) |
17 | 11, 12, 15, 16 | syl3anc 1439 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → (𝑇 ⊕ 𝑞) ∈ 𝑆) |
18 | 8 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑇𝐶𝑈) |
19 | simp3l 1215 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → 𝑇 ⊊ (𝑇 ⊕ 𝑞)) | |
20 | simp3r 1216 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → (𝑇 ⊕ 𝑞) ⊆ 𝑈) | |
21 | 1, 7, 11, 12, 13, 17, 18, 19, 20 | lcvnbtwn2 35181 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) → (𝑇 ⊕ 𝑞) = 𝑈) |
22 | 21 | 3exp 1109 | . . 3 ⊢ (𝜑 → (𝑞 ∈ 𝐴 → ((𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈) → (𝑇 ⊕ 𝑞) = 𝑈))) |
23 | 22 | reximdvai 3196 | . 2 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 (𝑇 ⊕ 𝑞) = 𝑈)) |
24 | 10, 23 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊕ 𝑞) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ⊆ wss 3792 ⊊ wpss 3793 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 LSSumclsm 18433 LModclmod 19255 LSubSpclss 19324 LSAtomsclsa 35128 ⋖L clcv 35172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lsatoms 35130 df-lcv 35173 |
This theorem is referenced by: islshpcv 35207 |
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